Finite Field

A Field with finitely many elements.

Lemma

A finite field has order of a prime power.
Moreover, there is a unique finite field of each prime power.

Proof

A finite field has prime Characteristic of a ring
(otherwise, it wouldn’t be an Integral Domain)

So say that our field has characteristic where is prime.
Then consider the field as the vector space of dimension over
This has size .

Corollary

A Freshman’s Dream holds in any finite field.

Lemma

For prime :

Theorem

Any finite field can be represented as a quotient

where is an irreducible polynomial of degree and is prime.
The field will have order .
So we can identify

Lemma

For for , consider the unique field .
Then the multiplicative group is cyclic.
We say that is a primitive element if